Analysis suggests that best score voting strategy is to give a max score to all candidates you like better than the expected value of the winner, and min-score the others. With approval voting, that just means "vote for the candidates you like better than the expected value of the winner."

Example

Say my sincere normalized preferences for the candidates in the 2008 Democratic presidential primary were:


Then I would assign probabilities to the winners (based on then-current polling and primary results) like:

If I wanted to be really unbiased, I could even look at the trading prices on web sites where people play the "candidate stock market" (i.e. gamble). But going with this rough estimate, I calculate my expected value from the winner to be:


So if I were maximally strategic, I would want to vote for every candidate I like more than 5.01, as follows:


With score voting, I would give these candidates a maximum score, and all the others a minimum score.

About the "singleton (plurality-style) voting" myth

It is totally false to claim that the best approval-style vote is always to vote for one candidate. It is the best way to vote if your utility for electing your favorite candidate, is tremendously higher than for every other. But that is generally not the case – and it is totally false to claim real-world voters will act that way. To prove that:

ANES polling data shows approximately 90% of the voters who claimed their favorite was Nader in 2000 strategically voted for someone else, suggesting that the vast majority of voters care more about maximizing their effect on the election than on getting their favorite elected at all costs.

No math skills?

If you're a little cynical about the math skills (or at least the math enjoyment) of the average voter, you're not alone. It is reasonable to expect that many Approval Voting users would mistakenly vote sub-optimally – especially when there are 3 or more strong candidates.

Now consider that a sincere score voting ballot has about 91% (or more) of the effectiveness of a perfectly strategic vote (in the 3-candidate study; 80% or more in the 5-candidate case), and it requires essentially no thought and no math.

So there's a case that a rational (but not necessarily mathematically inclined or ambitious) approval voting user often would rather just cast a sincere score voting ballot.

Sincere score voting can however be poor strategy in cases where there are two front-runners whose utilities are (in the voter's view) close together. But in that case a good strategy (which happens still to be fairly sincere) again is pretty obvious: a voter would be "safe" just to polarize the two front-runners and keep her other scores honest. For example, if in the above Clinton & Obama were regarded as the "front runners" with the only good winning chances, then a voter adopting the "polarization+honesty" strategy would vote maximum for Dodd, Gravel, and Obama; minimum for Clinton, and scaled in between for Richardson, Kucinich, Biden, and Edwards.

(Surprising) conclusion

Some critics have argued that score voting may trick unsuspecting naive voters into foolishly casting honest (and hence strategically weak) votes. But score voting can also actually help such voters by allowing them to cast a strategically fairly-good vote with no effort.

Certain people constantly criticize score voting as "just like taking approval voting and trying to dupe the naive into casting weak votes," which is "unfair." My point is, on the average, people quite-possibly are better off using score voting than using approval, if you consider how often people may use approval voting sub-optimally.